Section 85.15 (09WK): The site associated to a semi-representable object

85.15 The site associated to a semi-representable object

Let $\mathcal{C}$ be a site. Recall that a semi-representable object of $\mathcal{C}$ is simply a family $\{ U_ i\} _{i \in I}$ of objects of $\mathcal{C}$ . A morphism $\{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ of semi-representable objects is given by a map $\alpha : I \to J$ and for every $i \in I$ a morphism $f_ i : U_ i \to V_{\alpha (i)}$ of $\mathcal{C}$ . The category of semi-representable objects of $\mathcal{C}$ is denoted $\text{SR}(\mathcal{C})$ . See Hypercoverings, Definition 25.2.1 and the enclosing section for more information.

For a semi-representable object $K = \{ U_ i\} _{i \in I}$ of $\mathcal{C}$ we let

$\mathcal{C}/K = \coprod \nolimits _{i \in I} \mathcal{C}/U_ i$

be the disjoint union of the localizations of $\mathcal{C}$ at $U_ i$ . There is a natural structure of a site on this category, with coverings inherited from the localizations $\mathcal{C}/U_ i$ . The site $\mathcal{C}/K$ is called the localization of $\mathcal{C}$ at $K$ . Observe that a sheaf on $\mathcal{C}/K$ is the same thing as a family of sheaves $\mathcal{F}_ i$ on $\mathcal{C}/U_ i$ , i.e.,

$\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \nolimits _{i \in I} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$

This is occasionally useful to understand what is going on.

Let $\mathcal{C}$ be a site. Let $K = \{ U_ i\} _{i \in I}$ be an object of $\text{SR}(\mathcal{C})$ . There is a continuous and cocontinuous localization functor $j : \mathcal{C}/K \to \mathcal{C}$ which is the product of the localization functors $j_ i : \mathcal{C}/V_ i \to \mathcal{C}$ . We obtain functors $j_!$ , $j^{-1}$ , $j_*$ exactly as in Sites, Section 7.25. In terms of the product decomposition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \nolimits _{i \in I} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ we have

$\begin{matrix} j_! & : & (\mathcal{F}_ i)_{i \in I} & \longmapsto & \coprod j_{i, !}\mathcal{F}_ i \\ j^{-1} & : & \mathcal{G} & \longmapsto & (j_ i^{-1}\mathcal{G})_{i \in I} \\ j_* & : & (\mathcal{F}_ i)_{i \in I} & \longmapsto & \prod j_{i, *}\mathcal{F}_ i \end{matrix}$

as the reader easily verifies.

Let $f : K \to L$ be a morphism of $\text{SR}(\mathcal{C})$ . Then we obtain a continuous and cocontinuous functor

$v : \mathcal{C}/K \longrightarrow \mathcal{C}/L$

by applying the construction of Sites, Lemma 7.25.8 to the components. More precisely, suppose $f = (\alpha , f_ i)$ where $K = \{ U_ i\} _{i \in I}$ , $L = \{ V_ j\} _{j \in J}$ , $\alpha : I \to J$ , and $f_ i : U_ i \to V_{\alpha (i)}$ . Then the functor $v$ maps the component $\mathcal{C}/U_ i$ into the component $\mathcal{C}/V_{\alpha (i)}$ via the construction of the aforementioned lemma. In particular we obtain a morphism

$f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L)$

of topoi. In terms of the product decompositions $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \nolimits _{i \in I} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ and $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L) = \prod \nolimits _{j \in J} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V_ j)$ the reader verifies that

$\begin{matrix} f_! & : & (\mathcal{F}_ i)_{i \in I} & \longmapsto & (\coprod \nolimits _{i \in I, \alpha (i) = j} f_{i, !}\mathcal{F}_ i)_{j \in J} \\ f^{-1} & : & (\mathcal{G}_ j)_{j \in J} & \longmapsto & (f_ i^{-1}\mathcal{G}_{\alpha (i)})_{i \in I} \\ f_* & : & (\mathcal{F}_ i)_{i \in I} & \longmapsto & (\prod \nolimits _{i \in I, \alpha (i) = j} f_{i, *}\mathcal{F}_ i)_{j \in J} \end{matrix}$

where $f_ i : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V_{\alpha (i)})$ is the morphism associated to the localization functor $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ corresponding to $f_ i : U_ i \to V_{\alpha (i)}$ .

Lemma 85.15.1. Let $\mathcal{C}$ be a site.

For $K$ in $\text{SR}(\mathcal{C})$ the functor $j : \mathcal{C}/K \to \mathcal{C}$ is continuous, cocontinuous, and has property P of Sites, Remark 7.20.5.
For $f : K \to L$ in $\text{SR}(\mathcal{C})$ the functor $v : \mathcal{C}/K \to \mathcal{C}/L$ (see above) is continuous, cocontinuous, and has property P of Sites, Remark 7.20.5.

Proof. Proof of (2). In the notation of the discussion preceding the lemma, the localization functors $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ are continuous and cocontinuous by Sites, Section 7.25 and satisfy $P$ by Sites, Remark 7.25.11. It is formal to deduce $v$ is continuous and cocontinuous and has $P$ . We omit the details. We also omit the proof of (1). $\square$

Lemma 85.15.2. Let $\mathcal{C}$ be a site and $K$ in $\text{SR}(\mathcal{C})$ . For $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ we have

$j_*j^{-1}\mathcal{F} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (F(K)^\# , \mathcal{F})$

where $F$ is as in Hypercoverings, Definition 25.2.2.

Proof. Say $K = \{ U_ i\} _{i \in I}$ . Using the description of the functors $j^{-1}$ and $j_*$ given above we see that

$j_*j^{-1}\mathcal{F} = \prod \nolimits _{i \in I} j_{i, *}(\mathcal{F}|_{\mathcal{C}/U_ i}) = \prod \nolimits _{i \in I} \mathop{\mathcal{H}\! \mathit{om}}\nolimits (h_{U_ i}^\# , \mathcal{F})$

The second equality by Sites, Lemma 7.26.3. Since $F(K) = \coprod h_{U_ i}$ in $\textit{PSh}(\mathcal{C}$ , we have $F(K)^\# = \coprod h_{U_ i}^\#$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and since $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (-, \mathcal{F})$ turns coproducts into products (immediate from the construction in Sites, Section 7.26), we conclude. $\square$

Lemma 85.15.3. Let $\mathcal{C}$ be a site.

For $K$ in $\text{SR}(\mathcal{C})$ the functor $j_!$ gives an equivalence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\#$ where $F$ is as in Hypercoverings, Definition 25.2.2.
The functor $j^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ corresponds via the identification of (1) with $\mathcal{F} \mapsto (\mathcal{F} \times F(K)^\# \to F(K)^\# )$ .
For $f : K \to L$ in $\text{SR}(\mathcal{C})$ the functor $f^{-1}$ corresponds via the identifications of (1) to the functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(L)^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\#$ , $(\mathcal{G} \to F(L)^\# ) \mapsto (\mathcal{G} \times _{F(L)^\# } F(K)^\# \to F(K)^\# )$ .

Proof. Observe that if $K = \{ U_ i\} _{i \in I}$ then the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ decomposes as the product of the categories $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ . Observe that $F(K)^\# = \coprod _{i \in I} h_{U_ i}^\#$ (coproduct in sheaves). Hence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\#$ is the product of the categories $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_{U_ i}^\#$ . Thus (1) and (2) follow from the corresponding statements for each $i$ , see Sites, Lemmas 7.25.4 and 7.25.7. Similarly, if $L = \{ V_ j\} _{j \in J}$ and $f$ is given by $\alpha : I \to J$ and $f_ i : U_ i \to V_{\alpha (i)}$ , then we can apply Sites, Lemma 7.25.9 to each of the re-localization morphisms $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ to get (3). $\square$

Lemma 85.15.4. Let $\mathcal{C}$ be a site. For $K$ in $\text{SR}(\mathcal{C})$ the functor $j^{-1}$ sends injective abelian sheaves to injective abelian sheaves. Similarly, the functor $j^{-1}$ sends K-injective complexes of abelian sheaves to K-injective complexes of abelian sheaves.

Proof. The first statement is the natural generalization of Cohomology on Sites, Lemma 21.7.1 to semi-representable objects. In fact, it follows from this lemma by the product decomposition of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ and the description of the functor $j^{-1}$ given above. The second statement is the natural generalization of Cohomology on Sites, Lemma 21.20.1 and follows from it by the product decomposition of the topos.

Alternative: since $j$ induces a localization of topoi by Lemma 85.15.3 part (1) it also follows immediately from Cohomology on Sites, Lemmas 21.7.1 and 21.20.1 by enlarging the site; compare with the proof of Cohomology on Sites, Lemma 21.13.3 in the case of injective sheaves. $\square$

Remark 85.15.5 (Variant for over an object). Let $\mathcal{C}$ be a site. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ . The category $\text{SR}(\mathcal{C}, X)$ of semi-representable objects over $X$ is defined by the formula $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$ . See Hypercoverings, Definition 25.2.1. Thus we may apply the above discussion to the site $\mathcal{C}/X$ . Briefly, the constructions above give

a site $\mathcal{C}/K$ for $K$ in $\text{SR}(\mathcal{C}, X)$ ,
a decomposition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ if $K = \{ U_ i/X\}$ ,
a localization functor $j : \mathcal{C}/K \to \mathcal{C}/X$ ,
a morphism $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L)$ for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$ .

All results of this section hold in this situation by replacing $\mathcal{C}$ everywhere by $\mathcal{C}/X$ .

Remark 85.15.6 (Ringed variant). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings on $\mathcal{C}$ . In this case, for any semi-representable object $K$ of $\mathcal{C}$ the site $\mathcal{C}/K$ is a ringed site with sheaf of rings $\mathcal{O}_ K = j^{-1}\mathcal{O}_\mathcal {C}$ . The constructions above give

a ringed site $(\mathcal{C}/K, \mathcal{O}_ K)$ for $K$ in $\text{SR}(\mathcal{C})$ ,
a decomposition $\textit{Mod}(\mathcal{O}_ K) = \prod \textit{Mod}(\mathcal{O}_{U_ i})$ if $K = \{ U_ i\}$ ,
a localization morphism $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ of ringed topoi,
a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L), \mathcal{O}_ L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C})$ .

Many of the results above hold in this setting. For example, the functor $j^*$ has an exact left adjoint

$j_! : \textit{Mod}(\mathcal{O}_ K) \to \textit{Mod}(\mathcal{O}_\mathcal {C}),$

which in terms of the product decomposition given in (2) sends $(\mathcal{F}_ i)_{i \in I}$ to $\bigoplus j_{i, !}\mathcal{F}_ i$ . Similarly, given $f : K \to L$ as above, the functor $f^*$ has an exact left adjoint $f_! : \textit{Mod}(\mathcal{O}_ K) \to \textit{Mod}(\mathcal{O}_ L)$ . Thus the functors $j^*$ and $f^*$ are exact, i.e., $j$ and $f$ are flat morphisms of ringed topoi (also follows from the equalities $\mathcal{O}_ K = j^{-1}\mathcal{O}_\mathcal {C}$ and $\mathcal{O}_ K = f^{-1}\mathcal{O}_ L$ ).

Remark 85.15.7 (Ringed variant over an object). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings on $\mathcal{C}$ . Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and denote $\mathcal{O}_ X = \mathcal{O}_\mathcal {C}|_{\mathcal{C}/U}$ . Then we can combine the constructions given in Remarks 85.15.5 and 85.15.6 to get

a ringed site $(\mathcal{C}/K, \mathcal{O}_ K)$ for $K$ in $\text{SR}(\mathcal{C}, X)$ ,
a decomposition $\textit{Mod}(\mathcal{O}_ K) = \prod \textit{Mod}(\mathcal{O}_{U_ i})$ if $K = \{ U_ i\}$ ,
a localization morphism $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X)$ of ringed topoi,
a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L), \mathcal{O}_ L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$ .

Of course all of the results mentioned in Remark 85.15.6 hold in this setting as well.

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85.15 The site associated to a semi-representable object

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